Properties of ROC

The Region of Convergence of the Z-transform is not a secondary detail — it is what distinguishes between fundamentally different signals that share the same algebraic Z-transform expression. For GATE aspirants and DSP students, understanding ROC properties is directly linked to determining causality, stability, and the correct inverse Z-transform of any system.

Core Concept Explanation

The Region of Convergence (ROC) is the set of all complex values of z for which the Z-transform summation converges absolutely. Because the same rational Z-transform expression can correspond to multiple different time-domain sequences depending on the ROC chosen, the ROC is an inseparable part of any Z-transform specification.

The ROC is always a connected annular region in the z-plane: r₁ < |z| < r₂. The inner radius r₁ and outer radius r₂ are determined by the locations of poles. The ROC never contains any pole of X(z), because at a pole, the summation diverges. This is a fundamental rule used in virtually every ROC-related GATE question.

For a right-sided sequence (x[n] = 0 for n < N₀), the ROC extends outward from the outermost pole to infinity: |z| > |pmax|. For a causal sequence where N₀ = 0, the ROC always includes z = ∞. For a left-sided sequence (x[n] = 0 for n > N₀), the ROC extends inward from the innermost pole to the origin: |z| < |pmin|. For an anti-causal sequence where N₀ = 0, the ROC includes z = 0.

For a two-sided sequence that has nonzero values on both sides of the time axis, the ROC is an annular ring: r₁ < |z| < r₂. This ring lies strictly between two circles whose radii are determined by poles. If no such ring exists (i.e., poles are arranged such that no annulus can be formed free of poles), then no bilateral Z-transform exists for that sequence.

Mathematical Expression

The key ROC properties used in problem solving are: (1) ROC is an annular ring centered at the origin. (2) ROC does not contain any pole. (3) If x[n] is causal and rational, ROC is |z| > magnitude of largest pole. (4) If x[n] is anti-causal, ROC is |z| < magnitude of smallest pole. (5) If x[n] is finite duration (FIR), ROC is the entire z-plane except possibly z = 0 or z = ∞ depending on whether the sequence is one-sided or two-sided. (6) For a stable LTI system, the ROC of H(z) must include the unit circle |z| = 1. (7) For a causal stable system, all poles must lie strictly inside the unit circle.

The stability-causality connection is one of the most exam-relevant properties. A causal system has ROC: |z| > |pmax|. For this system to also be stable, |pmax| < 1 must hold, meaning all poles are inside the unit circle. If a system is stable but not causal, its ROC includes the unit circle but does not extend to infinity.

Practical Understanding

In practice, when designing an IIR digital filter, the designer places poles and zeros in the z-plane. The ROC for a causal filter is automatically the exterior of a circle passing through the outermost pole. Stability is then directly checked by verifying all poles lie inside the unit circle. This graphical approach is used in tools like MATLAB's zplane() function.

When inverting a Z-transform with multiple poles and no given ROC, engineers must consider all possible ROCs (each annular ring between consecutive poles) and determine which ROC corresponds to a physically realizable or stable system. This is why ROC knowledge is not just academic — it directly governs whether a designed filter will be stable or grow unbounded.

Given:
X(z) = z / [(z - 0.5)(z - 2)]
Find the ROC for: (a) causal sequence, (b) anti-causal sequence, (c) two-sided stable sequence.

Why this formula applies:
Poles at z = 0.5 and z = 2 define the boundaries of possible ROC regions.
Three annular rings are possible: |z| < 0.5, 0.5 < |z| < 2, |z| > 2.

Formula:
ROC cannot include any pole.
Causal → ROC extends to +∞ → |z| > 2
Anti-causal → ROC includes z=0 → |z| < 0.5
Two-sided stable → ROC must include unit circle |z|=1 → 0.5 < |z| < 2

Substitution:
|z| = 1 check: 0.5 < 1 < 2 → Yes, unit circle is inside the annulus.

Calculation:
For two-sided case, causal part uses pole at 0.5, anti-causal part uses pole at 2.

Final Answer:
(a) Causal: ROC = |z| > 2
(b) Anti-causal: ROC = |z| < 0.5
(c) Two-sided stable: ROC = 0.5 < |z| < 2  (includes unit circle)

Exam Tip: In GATE, whenever you see 'causal and stable system', immediately know that all poles must lie strictly inside the unit circle AND ROC = |z| > largest pole magnitude. These two conditions together are the complete answer.

• ROC never includes a pole. The presence of a pole means divergence at that z value.
• Right-sided sequence: ROC is exterior of a circle passing through the outermost pole.
• Left-sided sequence: ROC is interior of a circle passing through the innermost pole.
• Two-sided sequence: ROC is an annular ring. It must be checked against each possible ring.
• Stable system requires ROC to include the unit circle.
• Causal stable system: all poles inside unit circle, ROC = |z| > max pole magnitude.

Quick Revision

• ROC is always an annular ring: r₁ < |z| < r₂; never contains a pole.
• Causal ROC: |z| > outermost pole. Anti-causal ROC: |z| < innermost pole.
• Stability condition: ROC must include the unit circle |z| = 1.
• Causal + Stable: all poles strictly inside unit circle.
• Finite-duration sequence: ROC is entire z-plane (with possible exceptions at 0 or ∞).
• Trap: A causal system is not automatically stable. Check pole magnitudes separately.
• Trap: For two-sided X(z) with no stated ROC, there are multiple valid inverse Z-transforms — one per possible annular region.